Coherent control of an ultrabright single spin in hexagonal boron nitride at room temperature

Hexagonal boron nitride (hBN) is a remarkable two-dimensional (2D) material that hosts solid-state spins and has great potential to be used in quantum information applications, including quantum networks. However, in this application, both the optical and spin properties are crucial for single spins but have not yet been discovered simultaneously for hBN spins. Here, we realize an efficient method for arraying and isolating the single defects of hBN and use this method to discover a new spin defect with a high probability of 85%. This single defect exhibits outstanding optical properties and an optically controllable spin, as indicated by the observed significant Rabi oscillation and Hahn echo experiments at room temperature. First principles calculations indicate that complexes of carbon and oxygen dopants may be the origin of the single spin defects. This provides a possibility for further addressing spins that can be optically controlled.


I. SUPPLEMENTARY NOTE 1
The correction of the emission rate of Defect A. The saturated emission rate I sat measured in the experiment is ∼ 3.7 × 10 6 counts/s. The corrected emission rate I corrected sat can be written as I corrected sat = I sat /η, where η is the loss coefficient. The loss coefficient η is ∼ 14.9% (including the objective collection efficiency considering the reflection of the gold film (56%), fiber coupling effi- The room-temperature ODMR spectrum of Defect A at a 35.8-mT external magnetic field exhibits a positive peak at 970 MHz, whose contrast exceeds 2% and FWHM is ∼ 37 MHz.

II. SUPPLEMENTARY NOTE 2
A relatively higher ODMR contrast. Supplementary Fig. 1 shows a room-temperature ODMR spectrum with a contrast of more than 2% for Defect A. This value is obtained when the sample is first tested, and the difference may be induced by the variations in experimental conditions, especially the resistance that affects the microwave radiation efficiency. Nevertheless, this higher contrast exhibits the intrinsic ability of this defect.

III. SUPPLEMENTARY NOTE 3
Magnetic-field, laser-power and microwavepower-dependent ODMR of Defect A. Supplementary Fig. 2 shows the ODMR spectra at seven magnetic fields between 30.4∼36.1 mT at 100-µW laser power and 25-dBm microwave power, which exhibit a similar FWHM of the resonance. Supplementary Fig. 3 shows the ODMR spectra at different laser powers and microwave powers at a 34-mT magnetic field. The FWHM of the resonance at different powers suggests the power broadening phenomenon.
The corresponding FWHM values are shown in Fig. 5 of the main text.

IV. SUPPLEMENTARY NOTE 4
Part of the measurement results of the ultrabright color centers on the array sample. Supplementary Table 1  laser excitation at 532 nm, displaying a high probability of obtaining optically detectable single spins among the bright spots in the array. More importantly, these defects all exhibit consistent optical properties (ZPL at 540±10 nm) and ODMR signals. A type of previously reported single spin defect. In addition to the newly found ultrabright single spin defect described here, we also find a type of single spin defect that has been reported before [1,2], i.e., Defect J, as shown in Supplementary Fig. 4(a). The photoluminescence count of Defect J is ∼ 1 × 10 5 counts/s under 100-µW laser excitation, which is lower than Defect A (∼ 7 × 10 5 counts/s). Supplementary Fig. 4(b) shows the room-temperature PL emission spectrum of Defect J, revealing a ZPL at ∼ 576 nm and PSB at ∼ 588 nm. The second-order autocorrelation function g (2) (τ ) of De- fect J is presented in Supplementary Fig. 4(c), showing evidence of quantum emission but with background fluorescence contributions resulting in a g (2) (0) value of ∼ 0.60. Supplementary Fig. 4(d) displays the ODMR spectrum of Defect J under a 35-mT external magnetic field, exhibiting a positive contrast of ∼ 1% at 970 MHz. The optical and spin properties of Defect J are consistent with previously reported carbon-related defects [1,2]; thus, we can classify this type of defect as a carbon-related defect. We want to note that Defect J actually does not reach the 0.5-MHz brightness threshold; therefore, it is not in our list when we calculate the probabilities in Table II of the main text.

VI. SUPPLEMENTARY NOTE 6
Other novel defects. In addition to Defect A described in the main text, we identify two different new defects that exhibit a measurable ODMR signal and are obviously distinct from Defect A, and they are denoted as Defect K and Defect L in the following. Supplementary Figs. 5(a) and (d) display the photoluminescence spectra of Defect K and Defect L at room temperature.
The spectra are truncated at 540 nm due to the limitation of our 532-nm longpass filter. The ZPL of Defect K is below 540 nm. Defect L possesses a distinct ZPL at ∼ 546 nm and two relatively weak PSBs at ∼ 565 nm and ∼ 588 nm. We also show in Supplementary Figs. 5(b) and (e) that the second-order autocorrelation values at zero delay g (2) (0) for Defect K and Defect L are 0.53 and 0.39, respectively. Supplementary Figs. 5(c) and (f) show the ODMR spectra of Defect K and L at room temperature. The ODMR spectrum of Defect K exhibits six hyperfine lines and can be fitted well by a six-Lorentzian function. The results yield a hyperfine splitting constant of ∼ 50 MHz, which is similar to the hyperfine structure of V − B defects [3]. Defect L possesses a positive high contrast of 7% and a relatively wide FWHM of 116 MHz.

VII. SUPPLEMENTARY NOTE 7
The defect level diagram of the positively charged C B O N at C 2v symmetry is shown in Supplementary Fig.  6(a). There is an occupied b 2 state and empty b 2 and a 2 states in the spin majority channel (spin-up channel in the figure). The wavefunction shows an out-of-plane Time delay τ (μs) g (2) (τ) g (2) (τ) Microwave frequency (MHz) Microwave frequency (MHz)  spatial distribution, which is consistent with recent results from angular dependence analysis [4]. The transition dipole moment is oriented in plane. The hyperfine tensors are calculated for the nuclear spin active isotopes ( 13 C, 11 B and 14 N), as shown in Supplementary  Table 2. Although 13 C has an active nuclear spin, the natural abundance is low (1.1%) and, therefore, cannot Supplementary  single spin centers in the experiment. Replacing 11 B with 10 B further decreases the linewidth. The PSB simulation is based on the Franck-Condon approximation, which can be described by the overlap between the phonon mode in ground and excited states. Here, the calculation of the luminescence spectrum requires three steps: 1. computing the electronic ground and excited state with geometry optimization by HSE functionals, 2. computing the phonon in the electronic ground state by PBE functionals, and 3. computing the overlap between the phonon modes in the electronic ground and excited states. The temperature effect on phonon is not included. The maximum of the PSB is located at approximately 580 nm. The simulated PSB is larger than the experimental result. However, the C 2v symmetry is unstable, and the oxygen atom makes the carbon atom move out-of-plane. This might be due to the repulsion between C B and O N since they both act as donors in hBN. As a consequence, the calculated ZPL energies are lower (1.71 eV) and the Debye-Waller factor is higher in the low-symmetry configuration with respect to the C 2v configuration, as shown in Supplementary Fig.  7, and the calculated hyperfine constants are also affected considerably in the electronic ground state.
We consider the positively charged double C B (2C B ) and triple neutral C B (3C B ) and find that the ZPLs are smaller than the observed ZPL [9]. The ODMR spectra are quite similar to that of the single C B defect with a linewidth of approximately 50 MHz. Increasing the C B -C B distance in 2C B decreases the energy difference between two defect orbitals and decreases the ZPL energy. C 2 C B has an ODMR width of approximately 27 MHz, but the ZPL is less than the observed range [7]. Recently, C 2 C N has been considered as a possible single photon emitter with a ZPL at 2.33 eV for the second optical transition [8]. The simulated ODMR linewidth is 51 MHz. This might be related to Defect L in our study with respect to power broadening. Here, we cannot exclude the possibility of C N for the same reason. Unfortunately, this effect cannot currently be simulated. In addition to carbon-related defects, other configurations are also calculated. For intrinsic defects, the negatively charged boron antisite (B N ) has a ZPL at approximately 2.324 eV with S = 1.83; however, the FWHM of the ODMR broadening is 153 MHz. The positively charged nitrogen antisite (N B ) has a ZPL energy at 2.920 eV, which is out of the ballpark of the observed ZPL energies of single photon emitters in our study.
The gyromagnetic ratios are large for both boron and nitrogen isotopes, and their interactions with the electron spin generate a broad ODMR signal. The relatively narrow ODMR linewidth in the experiment is related to the spin density localization on either nuclear spin free isotopes with high natural abundance or nuclear spin isotopes with small gyromagnetic ratio, which is followed by the spin density overlapping with neighboring boron and nitrogen atoms with nonzero nuclear spins. The likely impurity candidates are carbon, oxygen and silicon atoms. Oxygen on the boron site (O B ) has extremely high formation energy, as indicated before, and oxygen on the nitrogen site (O N ) only has one occupied state close to CBM, which cannot be responsible for the observed transition [5]. Nor is silicon substitution a good candidate since its optical excitation energy is approximately 4 eV [6], and silicon is not a common defect without intentional doping. Complex pairs are also good candidates for quantum emitters. Complex pairs with native defects, including 7a (a is the lattice constant of hBN) has an ODMR linewidth of 119 MHz, and the optical signal has a ZPL of 2.30 eV with S = 2.38 HR factor. We conclude that this defect can also be a reasonable candidate for defect L. Further detailed investigation of donor-acceptor pairs (DAPs) might reveal the nature of the single spin species in hBN, but we focus on the most characterized Defect A in our study. In Supplementary Fig. 8, we plot the singly occupied Kohn-Sham wavefunction of several defects we consider here, which is mostly associated with the spin density distribution. C B is the defect we studied before, and the major contribution of hyperfine interactions comes from neighboring nitrogen atoms. Consequently, we speculate the narrow linewidth is due to the substitution of nitrogen with nuclear spin inactive elements discussed above. Removing one nitrogen creates the V N C B structure, which is a singlet. We then use hydrogen to passivate the carbon dangling bond, which is a doublet; however, hydrogen has a large hyperfine contribution. Thus, hydrogen is unlikely to exist in Defect A. The in-plane dangling bonds form an in-plane distribution wavefunction, which unavoidably interacts with the nuclear spin from boron and nitrogen, for example, in the negatively charged V N C B and C B O N V B defects. Therefore, in-plane dangling bonds should be avoided, which also means that the sp2 bonding type should be well preserved. Another example is C B O i where oxygen is an interlayer interstitial impurity. The carbon atom is pulled out of plane and transforms to the sp3 bonding type, which distributes the wavefunction on the nearby nitrogen atoms. The ODMR spectrum of C N O N -√ 7a is very similar to that of C N because the long distance between oxygen and carbon. Oxygen should be placed as the first neighbor of carbon to reduce hyperfine interactions from boron or nitride (C B O N ). The calculated hyperfine constants are listed in Supplementary Table 2. Indeed, the hyperfine constants of the immediate two nitrogen neighbors are moderate, resulting in a relatively narrow FWHM in the ODMR spectrum. However, as indicated above, there is strong repulsion between C B and O N since they both act as donors in hBN. Similarly, the C N and O B are unlikely to form a complex, as they are both acceptors. Alternatively, carbon can be used to substitute neighboring atoms to form carbon clusters. Here, we propose the C N C B3 defect, as plotted in Supplementary Fig. 9, in light of the ODMR spectra of C 2 C B . The positive charge state is a doublet and is stable when the Fermi level lies in the middle of the gap. The simulated ODMR linewidth is 28.5 MHz. However, the ZPL is 1.83 eV and S = 1.53. The calculated optical properties are not fully consistent with the high Debye-Waller factor and the ZPL energy at approximately 2.28 eV. Therefore, we conclude that these types of defects could be the core structures of more extended defects at which the spin density is localized, but their optical properties are perturbed by the presence of other nearby defects.

VIII. SUPPLEMENTARY NOTE 8
Structural characterization of hBN nanoparticles. Supplementary Fig. 11(a) shows the Raman spectrum of a nanoparticle with a bright spin defect on a Au substrate. The Raman spectrum exhibits a Raman shift of 1365.13 cm −1 , which is consistent with the previous Raman results of the hBN sample [10]. Supplementary  Fig. 11(b) presents a transmission electron microscope (TEM) image of hBN nanoparticles. The TEM image exhibits an evident lattice period with d (d-spacing) equal to 0.35 nm, which is marked by the red lines and arrows in Supplementary Fig. 11(b). The theoretical hBN lattice is also displayed in the inset of Supplementary Fig.  11(b), where the theoretical d is equal to 0.37 nm along the lattice orientation indicated by the blue lines in the inset. The similar d in the TEM image and the theoretical hBN lattice indicate that the nanoparticle containing the bright spin defect should be an hBN crystal. The slight difference between the theoretical lattice and our TEM results can be attributed to the lattice strain of the hBN sample or other reasons, such as temperature.  Fig. 8: Single spin wavefunction of proposed defects. Pure carbon-related defects can be found in Ref. [7][8][9] IX. SUPPLEMENTARY NOTE 9 Brightness statistics of the hBN defect arrays. In this work, we choose defects brighter than 0.5 MHz under 100-µW laser excitation as the reported bright spin defects, and the percentage of the hBN nanoparticles containing these bright defects with a single spin is approximately 0.1% among all hBN nanoparticles (regardless of bright or dark). Although the percentage of this bright defect is relatively low, we find that most of these bright defects (up to 85%) are isolated single spin defects with evident ODMR signals, and in addition, these bright spin defects can be easily selected from the PL map when the single hBN nanoparticles are arrayed; hence, it is an efficient method for finding the isolated spin defect in hBN nanoparticles by selecting the bright defects. As shown in Supplementary Fig. 12, there are mainly two categories of defects in our hBN nanoparticle sample (from the viewpoint of brightness), which exhibit very different brightness statistics. One category of nanoparticles exhibits very low brightness that is lower than 0.3 MHz, and the other one exhibits a much higher brightness that is higher than 0.8 MHz. Therefore, we chose the brightness threshold empirically as 0.5 MHz to select the bright spin defects in the same category.  Formation energy as a function of the Fermi level in N-rich and N-poor conditions. The stable 4C chain model here is for reference [9].There are 0, +1 and +2 stable charge states in the gap. Yellow and blue indicate the VB and CB of hBN.